Finite Group Databasebeta

← #36365203

Subgroups — #36365203

Order 22144.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149708C211yesmaximal, characteristic, complemented
2Q8×C22⋊D173C211yesmaximal, characteristic, complemented
3#26012140C211yesmaximal, characteristic
4#26011869C211yesmaximal, characteristic, complemented
5#26011749C211yesmaximal, characteristic, complemented
6#26011746C211yesmaximal, characteristic, complemented
7#26011699C211yesmaximal, characteristic, complemented
8#26011694C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11#26011647C211yesmaximal, characteristic, complemented
12#26011645C211yesmaximal, characteristic, complemented
13SD16×Dic173C211yesmaximal, characteristic, complemented
14#26011567C211yesmaximal, characteristic, complemented
15C2×#137×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18#19149708—24no—
19C2×Dic692C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21C2×#14583816C2211yescharacteristic
22C2×#14583816C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24C2×#14583809C2211yescharacteristic
25#19149677—21no—
26#19149677C2211yescharacteristic
27#19149677C2211yescharacteristic
28#19149662—21no—
29#19149660—21no—
30#19149658—21no—
31#19149658C2211yescharacteristic
32#19149658C2211yescharacteristic
33#19149655—21no—
34#19149655C2211yescharacteristic
35#19149655C2211yescharacteristic
36#19149651—21no—
37Q8×Dic173C2211yescharacteristic
38D4×Dic173C2211yescharacteristic
39C4×C22⋊D173—21no—
40C4×C22⋊D173C2211yescharacteristic
41C2×#14583805—21no—
42#19149611C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149603C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149600C2211yescharacteristic
50C22⋊#13063436C2211yescharacteristic
51#19149591C2211yescharacteristic
52C8×Dic173C2211yescharacteristic
53C2×#41×C173C2211yescharacteristic
54C22×Q8×C173C2211yescharacteristic
55#137×C173—24no—
56C2×Q16×C173C2211yescharacteristic
57C2×Q16×C173C2211yescharacteristic
58C2×SD16×C173C2211yescharacteristic
59C2×SD16×C173C2211yescharacteristic
60C2×M16×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—41no—
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173—41no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173D411yescharacteristic
70C22×Dic173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74Dic692—22no—
75#14583816—22no—
76#14583816—41no—
77#14583816—41no—
78#14583814—22no—
79#14583809—22no—
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91#14583805—21no—
92#14583805—21no—
93#14583805D411yescharacteristic
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97C22⋊Dic173D411yescharacteristic
98#14583803—21no—
99#14583803—21no—
100#14583803D411yescharacteristic
101C4×Dic173—21no—
102C4×Dic173C2311yescharacteristic
103#41×C173—22no—
104#41×C173—41no—
105C2×Q8×C173—21no—
106C2×Q8×C173—22no—
107C2×Q8×C173—41no—
108C2×Q8×C173C2311yescharacteristic
109C2×Q8×C173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116Q16×C173—22no—
117Q16×C173—22no—
118SD16×C173—22no—
119SD16×C173D414yescomplemented
120M16×C173—22no—
121C2×C8×C173C2311yescharacteristic
122C2×C8×C173C2311yescharacteristic
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—42no—
129C22⋊D173—22no—
130C22⋊D173—42no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—42no—
136C2×Dic173—42no—
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—42no—
143#13063436—21no—
144#13063436—21no—
145C23×C173—21no—
146C23×C173C2×D411yescharacteristic
147Q8×C173—21no—
148Q8×C173—21no—
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—42no—
153Q8×C173C2×D412yes—
154D4×C173—21no—
155D4×C173—42no—
156D4×C173C2×D412yes—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—42no—
163C2×C4×C173—42no—
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2411yescharacteristic
166C8×C173—21no—
167C8×C173C2×D412yes—
168C22×C173—21no—
169C22×C173—22no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173C22×D411yescharacteristic
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—21no—
178C4×C173—21no—
179C4×C173—22no—
180C4×C173—41no—
181C4×C173—41no—
182C4×C173C22×D411yescharacteristic
183C4×C173H(5,𝔽2)11yescharacteristic
184Dic173—21no—
185Dic173—21no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189Dic173—41no—
190Dic173#13712yescomplemented
191C2×C173—41no—
192C2×C173—42no—
193C2×C173D4211yescharacteristic
194C2×C173C2×#13711yescharacteristic
195C2×C173#57811yescharacteristic
196D173—41no—
197C173#327611yescharacteristic, complemented, Sylow
198#3276—1731nomaximal, Sylow
199C2×#137—1731no—
200C2×#137D17311yescharacteristic, complemented
201D4×Q8—1731no—
202#547—1731no—
203#494—1731no—
204#470—1731no—
205#470—1731no—
206#462—1731no—
207#461—1731no—
208#452—1731no—
209#452—1731no—
210#451—1731no—
211#451—1731no—
212C4×SD16—1731no—
213#436—1731no—
214C2×#41—1731no—
215C2×#41C2×D17311yescharacteristic
216C22×Q8—1731no—
217C22×Q8C2×D17311yescharacteristic
218#137—24no—
219#137—3464no—
220C2×Q16—1731no—
221C2×Q16—1731no—
222C2×Q16—1731no—
223C2×Q16—1731no—
224C2×Q16C2×D17311yescharacteristic
225C2×Q16C2×D17311yescharacteristic
226C2×SD16—1731no—
227C2×SD16C2×D17311yescharacteristic
228C2×SD16C2×D17311yescharacteristic
229C2×M16—1731no—
230C2×M16C2×D17311yescharacteristic
231C4⋊Q8—3461no—
232C4⋊Q8—1731no—
233C4⋊Q8—1731no—
234#123—3461no—
235#123—3461no—
236C22⋊Q8—3461no—
237C22⋊Q8—3461no—
238C22⋊Q8—1731no—
239C22⋊Q8—1731no—
240C22⋊Q8—1731no—
241C22⋊Q8—1731no—
242#121—3461no—
243C4×Q8—1731no—
244C4×D4—3461no—
245C4×D4—1731no—
246C4×D4—1731no—
247C2×C4⋊C4—3461no—
248#106—1731no—
249C4⋊C8—1731no—
250#103—1731no—
251#103—1731no—
252#103—1731no—
253#103—1731no—
254#103—1731no—
255#102—1731no—
256C22⋊C8—1731no—
257C22⋊C8—1731no—
258C4×C8—1731no—
259#41—22no—
260#41—41no—
261#41—3462no—
262#41—6921no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—3461no—
267C2×Q8—3462no—
268C2×Q8—1731no—
269C2×Q8—1731no—
270C2×Q8—6921no—
271C2×Q8C22×D17311yescharacteristic
272C2×Q8C22×D17311yescharacteristic
273C2×Q8C22×D17311yescharacteristic
274C2×D4—21no—
275C2×D4—3461no—
276C2×D4—1731no—
277C2×D4C22×D17311yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4—1731no—
284C22×C4C22×D17311yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—3462no—
288Q16—3462no—
289Q16—6921no—
290Q16—6921no—
291SD16—22no—
292SD16—3462no—
293SD16C22⋊D17314yescomplemented
294M16—22no—
295M16—3462no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8C22×D17311yescharacteristic
299C2×C8C22×D17311yescharacteristic
300C4⋊C4—3461no—
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—3461no—
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—1731no—
310C4⋊C4—1731no—
311C4⋊C4—1731no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—1731no—
318C22⋊C4—1731no—
319C42—3461no—
320C42—1731no—
321C23—21no—
322C23—1731no—
323C23D4×D17311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—3461no—
331Q8—3461no—
332Q8—3461no—
333Q8—3461no—
334Q8—6922no—
335Q8C2×C22⋊D17312yes—
336D4—21no—
337D4—42no—
338D4—3462no—
339D4—6922no—
340D4C2×C22⋊D17312yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—1731no—
355C2×C4—1731no—
356C2×C4—1731no—
357C2×C4—6922no—
358C2×C4—6922no—
359C2×C4—6922no—
360C2×C4D4×D17311yescharacteristic
361C2×C4C23×D17311yescharacteristic
362C8—21no—
363C8—3461no—
364C8—3461no—
365C8C2×C22⋊D17312yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—3461no—
373C22—3461no—
374C22—3461no—
375C22C2×D4×D17311yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—3461no—
381C4—3461no—
382C4—1732no—
383C4—6921no—
384C4—6921no—
385C4—6921no—
386C4—6921no—
387C4C22×C22⋊D17311yescharacteristic
388C4#1914972411yescharacteristic
389C2—41no—
390C2—42no—
391C2—6921no—
392C2D4×C22⋊D17311yescharacteristic
393C2#137×D17311yescharacteristic
394C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#137×C173C2
socleC22×C173C22×D4
prime radical#26012140C2
hypercenterC2×#137D173
hypocenterC173#3276
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
2. Q8×C22⋊D173C2semidirect
4. #26011869C2semidirect
5. #26011749C2semidirect
6. #26011746C2semidirect
7. #26011699C2semidirect
8. #26011694C2semidirect
9. #26011653C2semidirect
10. #26011650C2semidirect
11. #26011647C2semidirect
12. #26011645C2semidirect
13. SD16×Dic173C2semidirect
14. #26011567C2semidirect
15. C2×#137×C173C2semidirect
119. SD16×C173D4semidirect
190. Dic173#137semidirect
197. C173#3276semidirect
200. C2×#137D173semidirect
293. SD16C22⋊D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191497082
Q8×C22⋊D1732
#260121402
#260118692
#260117492
#260117462
#260116992
#260116942
#260116532
#260116502
#260116472
#260116452
SD16×Dic1732
#260115672
C2×#137×C1732
#3276173